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Journal of Applied Mathematics

Perturbation approach for nuclear magnetic resonance solid‐state quantum computation

G. P. Berman, D. I. Kamenev, V. I. Tsifrinovich

Published Jan 1, 2003• 2 citations
Publisher

Abstract

A dynamics of a nuclear‐spin quantum computer with a large number (L = 1000) of qubits is considered using a perturbation approach. Small parameters are introduced and used to compute the error in an implementation of an entanglement between remote qubits, using a sequence of radio‐frequency pulses. The error is computed up to the different orders of the perturbation theory and tested using exact numerical solution.

DOI
10.1155/s1110757x03110182

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A dynamics of a nuclear‐spin quantum computer with a large number (L = 1000) of qubits is considered using a perturbation approach. Small parameters are introduced and used to compute the error in an implementation of an entanglement between remote qubits, using a sequence of radio‐frequency pulses. The error is computed up to the different orders of the perturbation theory and tested using exact numerical solution.
The perturbation theory is developed based on small parameters which naturally appear in solid state quantum computation. We report the simulations of the dynamics of quantum logic operations with a large number of qubits (up to 1000). A nuclear spin chain is considered in which selective excitations of spins are provided by having a uniform gradient of the external magnetic field. Quantum logic operations are utilized by applying resonant electromagnetic pulses. The spins interact with their nearest neighbors. We simulate the creation of the long-distance entanglement between remote qubits in the spin chain. Our method enables us to minimize unwanted non-resonant effects in a controlled way. The method we use cannot simulate complicated quantum logic (a quantum computer is required to do this), but it can be useful to test the experimental performance of simple quantum logic operations. We show that: (a) the probability distribution of unwanted states has a ``band'' structure, (b) the directions of spins in typical unwanted states are highly correlated, and (c) many of the unwanted states are high-energy states of a quantum computer (a spin chain). Our approach can be applied to simple quantum logic gates and fragments of quantum algorithms involving a large number of qubits.
We develop a dynamical perturbation theory for solid-state quantum computation with many qubits. The errors in the implementation of simple quantum logic operations caused by nonresonant transitions are estimated. We verify our perturbation approach using exact numerical solution for relatively small (N=10)(N=10)(N=10) number of qubits. A preferred range of parameters is found in which the errors in processing quantum information are reasonably small. Our results can be useful for understanding the mechanisms of errors and for experimental testing of scalable solid-state quantum computers.
In this paper we analyze the dynamics in a spin model of quantum computer. Main attention is paid to the dynamical fidelity (associated with dynamical errors) of an algorithm that allows to create an entangled state for remote qubits. We show that in the regime of selective resonant excitations of qubits there is no danger of quantum chaos. Moreover, in this regime a modified perturbation theory gives an adequate description of the dynamics of the system. Our approach allows us to explicitly describe all peculiarities of the evolution of the system under time-dependent pulses corresponding to a quantum protocol. Specifically, we analyze, both analytically and numerically, how the fidelity decreases in dependence on the model parameters.
Nuclear magnetic resonance offers an appealing prospect for implementation of quantum computers, because of the long coherence times associated with nuclear spins, and extensive laboratory experience in manipulating the spins with radio frequency pulses. Existing proposals, however, suffer from a signal-to-noise ratio that decays exponentially in the number of qubits in the quantum computer. This places a severe limit on the size of the computations that can be performed by such a computer; estimates of that limit are well within the range in which a conventional computer taking exponentially more steps would still be practical. We give an NMR implementation in which the signal-to-noise ratio depends only on features of NMR technology, not the size of the computer. This provides a means for NMR computation techniques to scale to sizes at which the exponential speedup enables quantum computation to solve problems beyond the capabilities of classical computers.